Using error vector magnitude to estimate sample clock frequency offset

ABSTRACT

A low complexity system and method for operating a receiver in order to estimate an offset between the actual sample clock rate 1/T S ′ of a receiver and an intended sample clock rate 1/T S . The receiver captures samples of a received baseband signal at the rate 1/T S ′, operates on the captured samples to generate an estimate for the clock rate offset, and fractionally resamples the captured samples using the clock rate offset. The resampled data represents an estimate of baseband symbols transmitted by the transmitter. The action of operating on the captured samples involves computing an error vector signal and then estimating the clock rate offset using the error vector signal. The error vector signal may be computed in different ways depending on whether or not carrier frequency offset and carrier phase offset are assumed to be present in the received baseband signal.

CONTINUATION DATA

The present application is a continuation of U.S. patent applicationSer. No. 13/027,885, filed on Feb. 15, 2011, entitled Estimation ofSample Clock Frequency Offset Using Error Vector Magnitude”, which ishereby incorporated by reference in its entirety as though fully andcompletely set forth herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to the field of digital signal processing,and more specifically, to systems and methods for estimating the offsetbetween the actual sample-clock-frequency used by a receiver'sanalog-to-digital converter and an intended sample-clock-frequency.

2. Description of the Related Art

At a transmitter, a stream of information bits is mapped into a streamof complex symbols {U_(n)} in a constellation (i.e., a set of points inthe complex plane), e.g., a QAM constellation. The stream of symbols{U_(n)} may be supplied to a subsystem that converts the symbol streaminto an analog baseband signal u(t). The analog baseband signal u(t) maybe used to modulate a carrier signal. The modulated carrier istransmitted through a channel, e.g., a wired channel such as a cable, awireless channel such as the atmosphere or free space, a fiber opticchannel, etc. A receiver captures a channel-distorted version of thetransmitted signal from the channel, recovers a baseband signal from thecaptured signal, and samples the baseband signal. However, the samplingclock rate 1/T_(S)′ used by the receiver is in general different from anintended sampling clock rate 1/T_(S). This offset between the intendedclock rate 1/T_(S) and the actual clock rate 1/T_(S)′ makes it difficultto accurately demodulate the received baseband signal. Thus, thereexists a need for methods, especially low-complexity methods, forestimating this clock rate offset.

SUMMARY

A method for operating a receiver system according to the presentinvention may involve the following operations.

The method involves an analog-to-digital converter (ADC) of the receiversystem capturing samples {x(nT_(S)′)} of a received baseband signal,where n is a sample index, where T_(S)′ is a period of the sample clockof the ADC. The period T_(S)′ is not known at least initially. However,it may be estimated as one of the possible results of the presentmethod.

The method also involves a processor of the receiver system computing areference signal {r(n)} by performing carrier-frequency-offset (CFO)removal and carrier-phase-offset (CPO) removal on the baseband signalsamples {x(nT_(S)′)} to obtain corrected samples; by performing harddecision demodulation on the corrected samples to obtain a bit stream;and by re-modulating the bit stream to determine the reference signal.

The method also involves the processor computing an intermediate signal{Y(n)} by operating on the baseband signal samples {x(nT_(S)′)} toremove CFO and CPO using the reference signal {r(n)}.

The method also involves the processor computing an error vector signal{e(n)} based on the intermediate signal {Y(n)} and the reference signal{r(n)}, e.g., according to the relation: e(n)=Y(n)−r(n).

The method also involves the processor computing an estimate Q for theabsolute value |ε| of an offset parameter ε based on an average slope ofan absolute value of the error vector signal {e(n)}. The offsetparameter ε represents a relative offset between the period T_(S)′ andan intended period T_(S) of the ADC's sample clock.

The method also involves the processor fractionally resampling thebaseband signal samples {x(nT_(S)′)} by a factor (1+Q) to obtain a firstset of adjusted samples {Z₁(n)} corresponding to the sample ratef₁=(1+Q)/T_(S)′, and fractionally resampling the baseband signal samples{x(nT_(S)′)} by the factor (1−Q) to obtain a second set of adjustedsamples {Z₂(n)} corresponding to the sample rate f₂=(1−Q)/T_(S)′.

The method also involves the processor computing a first error vectorsignal {e₁(n)} based on the first set of adjusted samples {Z₁(n)} andthe reference signal {r(n)}, e.g., according to the expression:e₁(n)=Z₁(n)−r(n).

The method also involves the processor computing a second error vectorsignal {e₂(n)} based on the second set of adjusted samples {Z₂(n)} andthe reference signal {r(n)}, e.g., according to the expression:e₂(n)=Z₂(n)−r(n).

The method also involves computing an RMS value R₁ of the first errorvector signal {e₁(n)}, and computing an RMS value R₂ of the second errorvector signal {e₂(n)}.

The method also involves selecting either Q or −Q as an estimate for theoffset parameter ε based on whether or not R₁ is smaller than R₂. Theestimated value of ε may be used in any of a variety of ways. Forexample, it may be used to resample to received signal samples to theintended sample rate, to adjust the frequency of the ADC's sample clock,to compute an estimate for the period T_(S)′, etc.

Another method for operating a receiver system may involve the followingoperations.

The method involves an analog-to-digital converter (ADC) of the receiversystem capturing samples {x(nT_(S)′)} of a received baseband signal,where n is a sample index, where T_(S)′ is a period of a sample clock ofthe ADC. The period T_(S)′ is not known at least initially. However, itmay be estimated as one of the possible results of the present method.

The method also involves a processor of the receiver system computing areference signal {r(n)} by performing carrier-frequency-offset (CFO)removal and carrier-phase-offset (CPO) removal on the baseband signalsamples {x(nT_(S)′)} to obtain corrected samples; by performing harddecision demodulation on the corrected samples to obtain a bit stream;and by re-modulating the bit stream to determine the reference signal.

The method also involves the processor computing an intermediate signal{Y(n)} by operating on the baseband signal samples {x(nT_(S)′)} toremove the CFO and the CPO using the reference signal {r(n)}.

The method also involves the processor computing a differential of thereference signal {r(n)} and a differential of the intermediate signal{Y(n)}.

The method also involves the processor computing an error vector signal{e′(n)} based on the differential of the intermediate signal {Y(n)} andthe differential of the reference signal {r(n)}.

The method also involves the processor computing an estimate Q for theabsolute value |ε| of an offset parameter ε based on an RMS value of theerror vector signal {e′(n)}. The offset parameter ε represents arelative offset between the period T_(S)′ and an intended period T_(S)of the sample clock of the ADC.

The method also involves the processor fractionally resampling thebaseband signal samples {x(nT_(S)′)} by a factor of (1+Q) to obtain afirst set of adjusted samples {Z₁(n)} corresponding to sample ratef₁=(1+Q)/T_(S)′, and fractionally resampling the baseband signal samples{x(nT_(S)′)} by a factor of (1−Q) to obtain a second set of adjustedsamples {Z₂(n)} corresponding to sample rate f₂=(1−Q)/T_(S)′.

The method also involves the processor computing a first error vectorsignal {e₁(n)} based on the first set of adjusted samples {Z₁(n)} andthe reference signal {r(n)}, and computing a second error vector signal{e₂(n)} based on the second set of adjusted samples {Z₂(n)} and thereference signal {r(n)}.

The method also involves the processor selecting either Q or −Q as anestimate for the offset parameter ε, where the selection of either Q or−Q is based on whether or not an RMS value R₁ of the first error vectorsignal is smaller than an RMS value R₂ of the second error vectorsignal. The estimated value of ε may be used in any of a variety ofways. For example, it may be used to resample to received signal samplesto the intended sample rate 1/T_(S), to make corrective adjustments tothe frequency of the ADC's sample clock, to compute an estimate for thesample period T_(S)′, etc.

BRIEF DESCRIPTION OF THE DRAWINGS

A better understanding of the present invention can be obtained when thefollowing detailed description of the preferred embodiment is consideredin conjunction with the following drawings.

FIG. 1 illustrates the linearity of the phase α(t) between sampling timenT_(S) and sampling time nT_(S)′=(1+ε)nT_(S).

FIG. 2 illustrates one embodiment of a method for operating a receiversystem.

FIG. 3 illustrates one embodiment of a second method for operating areceiver system.

FIG. 4 illustrates one embodiment of a simulation framework used tosimulate the methods of FIG. 3 and FIG. 4.

FIG. 5 is a table showing one configuration that was used for simulationwithin the framework of FIG. 4.

FIG. 6 is a table showing the clock frequency offset (f_(S)′−f_(S))introduced by simulation, and the corresponding estimated value of theclock frequency offset.

FIG. 7 is a graph showing the actual sample clock rate introduced intothe baseband signal vs. iteration number, and the estimated sample clockrate vs. iteration number for the configuration given in FIG. 5.

FIGS. 8A and 8B represent the baseband 64QAM signal constellation before(FIG. 8A) and after (FIG. 8A) correction of the clock frequency offsetusing the method of FIG. 2.

FIGS. 9A and 9B shows the plot of the EVM function |e(n)| respectivelyfor clock frequency offset=1149 Hz and clock frequency offset=−326 Hz,but without CFO or CPO impairments.

FIG. 10 illustrates one embodiment of a computer system that may be usedto perform any of the method embodiments described herein.

FIG. 11 illustrates one possible embodiment of the computer system ofFIG. 10.

While the invention is susceptible to various modifications andalternative forms, specific embodiments thereof are shown by way ofexample in the drawings and will herein be described in detail. Itshould be understood, however, that the drawings and detaileddescription thereto are not intended to limit the invention to theparticular form disclosed, but on the contrary, the intention is tocover all modifications, equivalents, and alternatives falling withinthe spirit and scope of the present invention as defined by the appendedclaims. It is noted that the word “may” is used throughout thisapplication in a permissive sense (e.g., having the potential to, beingable to), not a mandatory sense (e.g., must).

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following description, numerous specific details are set forth toprovide a thorough understanding of embodiments of the presentinvention. However, one having ordinary skill in the art shouldrecognize that the invention may be practiced with various modificationsto these specific details.

TERMINOLOGY

The following is a glossary of terms used in the present document.

Memory Medium—A memory medium is a medium configured for the storage andretrieval of information. Examples of memory media include: variouskinds of semiconductor memory such as RAM and ROM; various kinds ofmagnetic media such as magnetic disk, tape, strip and film; variouskinds of optical media such as CD-ROM and DVD-ROM; various media basedon the storage of electrical charge and/or other physical quantities;media fabricated using various lithographic techniques; etc. The term“memory medium” may also include a set of two or more memory media whichreside at different locations, e.g., at different computers that areconnected over a network.

Programmable Hardware Element—a hardware device that includes multipleprogrammable function blocks connected via a programmable interconnect.Examples include FPGAs (Field Programmable Gate Arrays), PLDs(Programmable Logic Devices), FPOAs (Field Programmable Object Arrays),and CPLDs (Complex PLDs). The programmable function blocks may rangefrom fine grained (combinatorial logic or look up tables) to coarsegrained (arithmetic logic units or processor cores). A programmablehardware element may also be referred to as “reconfigurable logic”.

Program—the term “program” is intended to have the full breadth of itsordinary meaning. As used herein, the term “program” includes within itsscope of meaning: 1) a software program which is stored in a memory andis executable by a processor, or, 2) a hardware configuration programuseable for configuring a programmable hardware element. Any of themethod embodiments described herein, or, any combination of the methodembodiments described herein, or, any subset of any of the methodembodiments described herein, or, any combination of such subsets may beimplemented in terms of one or more programs.

Software Program—the term “software program” is intended to have thefull breadth of its ordinary meaning, and includes any type of programinstructions, code, script and/or data, or combinations thereof, thatmay be stored in a memory medium and executed by a processor or computersystem. Exemplary software programs include: programs written intext-based programming languages such as C, C++, Java™, Pascal, Fortran,Perl, etc.; graphical programs (programs written in graphicalprogramming languages); assembly language programs; programs that havebeen compiled to machine language; scripts; and other types ofexecutable software. A software program may comprise two or moresubprograms that interoperate in a specified manner.

Hardware Configuration Program—a program, e.g., a netlist or bit file,that can be used to program or configure a programmable hardwareelement.

Graphical Program—A program comprising a plurality of interconnectednodes or icons, where the plurality of interconnected nodes or iconsvisually indicate the functionality of the program. A graphical programis a type of diagram.

The following provides examples of various aspects of graphicalprograms. The following examples and discussion are not intended tolimit the above definition of graphical program, but rather provideexamples of what the term “graphical program” encompasses.

The nodes in a graphical program may be connected in one or more of adata flow, control flow, and/or, execution flow format. The nodes mayalso be connected in a “signal flow” format, which is a subset of dataflow.

Exemplary graphical program development environments which may be usedto create graphical programs include LabVIEW, DasyLab, DiaDem andMatrixx/SystemBuild from National Instruments, Simulink from theMathWorks, VEE from Agilent, WiT from Coreco, Vision Program Managerfrom PPT Vision, SoftWIRE from Measurement Computing, Sanscript fromNorthwoods Software, Khoros from Khoral Research, SnapMaster from HEMData, VisSim from Visual Solutions, ObjectBench by SES (Scientific andEngineering Software), and VisiDAQ from Advantech, among others.

The term “graphical program” includes models or block diagrams createdin graphical modeling environments, where the model or block diagramcomprises interconnected nodes or icons that visually indicate operationof the model or block diagram; exemplary graphical modeling environmentsinclude Simulink, SystemBuild, VisSim, Hypersignal Block Diagram, etc.

A graphical program may be represented in the memory of the computersystem as data structures and/or program instructions. The graphicalprogram, e.g., these data structures and/or program instructions, may becompiled or interpreted to produce machine language that accomplishesthe desired method or process as shown in the graphical program.

Input data to a graphical program may be received from any of varioussources, such as a receiver (e.g., an RF receiver) or a receiver frontend, a signal processing board, a modem, a network interface (e.g., awireless network interface), a unit under test, a process being measuredor controlled, another computer program, a database, or from a file.Also, a user may input data to a graphical program or virtual instrumentusing a graphical user interface, e.g., a front panel.

A graphical program may optionally have a GUI associated with thegraphical program. In this case, the plurality of interconnected nodesare often referred to as the block diagram portion of the graphicalprogram.

Data Flow Graphical Program (or Data Flow Diagram)—A graphical programor diagram comprising a plurality of interconnected nodes, where theconnections between the nodes indicate that data produced by one node isused by another node.

Node—In the context of a graphical program, an element that may beincluded in a graphical program. A node may have an associated icon thatrepresents the node in the graphical program, as well as underlying codeand/or data that implements functionality of the node. Exemplary nodesinclude function nodes, sub-program nodes (sub-VIs), terminal nodes,structure nodes, etc. Nodes may be connected together in a graphicalprogram by connection icons or wires.

Graphical User Interface—this term is intended to have the full breadthof its ordinary meaning. The term “Graphical User Interface” is oftenabbreviated to “GUI”. A GUI may include one or more input GUI elements,one or more output GUI elements, or both input and output GUI elements.

The following provides examples of various aspects of GUIs. Thefollowing examples and discussion are not intended to limit the ordinarymeaning of GUI, but rather provide examples of what the term “graphicaluser interface” encompasses.

A GUI may comprise a single window having one or more GUI elements, ormay comprise more than one window, each having one or more GUI Elements.

A GUI may be associated with a diagram, e.g., a graphical program. Inthis instance, various mechanisms may be used to connect GUI elements inthe GUI with nodes or icons in the diagram/graphical program. Forexample, when Input Controls and Output Indicators are created in theGUI, corresponding nodes (e.g., terminals) may be automatically createdin the diagram or graphical program. Alternatively, the user can placeterminal nodes in the diagram which may cause the display ofcorresponding GUI elements front panel objects in the GUI, either atedit time or later at run time. As another example, the GUI may compriseGUI Elements embedded in the block diagram portion of the graphicalprogram.

Front Panel—A Graphical User Interface that includes input controls andoutput indicators, and that enables a user to interactively control ormanipulate the input being provided to a program or diagram, and viewoutput of the program or diagram, during execution.

A front panel is a type of GUI. A front panel may be associated with adiagram or graphical program as described above.

In an instrumentation application, the front panel can be analogized tothe front panel of an instrument. In an industrial automationapplication the front panel can be analogized to the MMI (Man MachineInterface) of a device. The user may adjust the controls on the frontpanel to affect the input, and view the output on the respectiveindicators.

Graphical User Interface Element—an element of a graphical userinterface, such as for providing input or displaying output. Exemplarygraphical user interface elements comprise input controls and outputindicators

Input Control—a graphical user interface element for providing userinput to a program. Exemplary input controls comprise dials, knobs,sliders, switches, text input boxes, numeric input fields, etc.

Output Indicator—a graphical user interface element for displayingoutput from a program. Exemplary output indicators include charts,graphs, gauges, text output boxes, numeric displays, etc. An outputindicator is sometimes referred to as an “output control”.

Computer System—any of various types of computing or processing systems,including a personal computer (PC), a mainframe computer system, aworkstation, a laptop, a network appliance, an Internet appliance, ahand-held or mobile device, a personal digital assistant (PDA), atelevision system, a grid computing system, or other device orcombinations of devices. In general, the term “computer system” can bebroadly defined to encompass any device (or combination of devices)having at least one processor that is configured to execute instructionsthat are stored on a memory medium.

Measurement Device—includes instruments, data acquisition devices, smartsensors and any of various types of devices that are operable to acquireand/or store data. A measurement device may also optionally be furtheroperable to analyze or process the acquired or stored data. Examples ofa measurement device include an instrument, such as a traditionalstand-alone “box” instrument, a computer-based instrument (instrument ona card) or external instrument, a data acquisition card, a deviceexternal to a computer that operates similarly to a data acquisitioncard, a smart sensor, one or more DAQ or measurement cards or modules ina chassis, an image acquisition device, such as an image acquisition (ormachine vision) card, a video capture board, a smart camera, a motioncontrol device, a robot having machine vision, and other similar typesof devices. Exemplary “stand-alone” instruments include oscilloscopes,multimeters, signal analyzers, signal demodulators, arbitrary waveformgenerators, spectroscopes, and similar measurement, test, or automationinstruments.

A measurement device may be further operable to perform controlfunctions, e.g., in response to analysis of the acquired or stored data.For example, the measurement device may send a control signal to anexternal system, such as a motion control system or to a sensor, inresponse to particular data. A measurement device may also be operableto perform automation functions, e.g., may receive and analyze data, andissue automation control signals in response.

Embodiments of the present inventions may be realized in any of variousforms. For example, any of the present inventions may be realized as acomputer-implemented method, a computer-readable memory medium, or acomputer system. Furthermore, any of the present inventions may berealized using one or more custom-designed hardware devices such asASICs.

A computer-readable memory medium is a memory medium that stores programinstructions and/or data, where the program instructions, if executed bya computer system, cause the computer system to perform a method, e.g.,any of a method embodiments described herein, or, any combination of themethod embodiments described herein, or, any subset of any of the methodembodiments described herein, or, any combination of such subsets.

In some embodiments, a computer system may include a processor (or a setof interconnected processors) and a memory medium. The memory mediumstores program instructions. The processor is configured to read andexecute the program instructions from the memory medium. The programinstructions are executable by the processor to implement a method,e.g., any of the various method embodiments described herein (or, anycombination of the method embodiments described herein, or, any subsetof any of the method embodiments described herein, or, any combinationof such subsets). The computer system may be realized in any of variousforms. For example, the computer system may be a personal computer (inany of its various realizations), a workstation, a computer on a card,an application-specific computer in a box, a server computer, a clientcomputer, a hand-held device, a mobile device, a tablet computer, awearable computer, a computer integrated in a head-mounted display, etc.

In some embodiments, a set of two or more computers distributed across acomputer network may be configured to partition the effort of executinga computational method, e.g., any of the various method embodimentsdescribed herein (or, any combination of the method embodimentsdescribed herein, or, any subset of any of the method embodimentsdescribed herein, or, any combination of such subsets).

In some embodiments, a first computer may be configured to receive amodulated signal, down-convert the modulated signal to baseband, andcapture samples of the baseband signal. The first computer may send thecaptured samples to a second computer through the computer network. Thesecond computer may operate on the signal samples according to any ofthe method embodiments described herein, or, any combination of themethod embodiments described herein, or, any subset of any of the methodembodiments described herein, or, any combination of such subsets.

The following is a list of acronyms used herein.

BER Bit Error Rate

CFO Carrier Frequency offset

CPO Carrier phase offset

EVM Error Vector Magnitude

FPGA Field Programmable Gate Array

PLL Phase-Locked Loop

QAM Quadrature Amplitude Modulation

Clock frequency offset is the difference between the clock frequencyf_(S) of the digital-to-analog converter (DAC) at the transmitter andthe clock frequency f_(S)′ of the analog-to-digital converter (ADC) atthe receiver. Similarly, clock period offset is the difference betweenthe clock period T_(S) of the transmitter's DAC and the clock periodT_(S)′ of the receiver's ADC. Relative clock period offset ε is definedas

$\begin{matrix}{ɛ = {\frac{T_{S}^{\prime} - T_{S}}{T_{S}}.}} & (1)\end{matrix}$This patent describes among other things methods for estimating theparameter ε.

Under the assumption that the transmitter's sample clock conformsideally to an intended value of the period T_(S), e.g., a valuespecified by (or derived from) a communication protocol, the parameter εmay be interpreted as a measure of the departure of the period T_(S)′from the intended period value.

In some embodiments, the receiver may use the estimated value of theparameter ε to resample the samples captured by the ADC, i.e., toresample the captured samples to the intended sample rate 1/T_(S).

In some embodiments, the receiver may use the estimated value of theparameter ε to adjust the clock frequency f_(S)′ of the ADC in order todrive the clock frequency f_(S)′ towards the intended clock frequency1/T_(S).

In some embodiments, the receiver may compute an estimate for the periodT_(S)′ based on the estimated value of the parameter ε and the intendedperiod value T_(S). The computed value of the period T_(S)′ may bereported/displayed to a user, e.g., via a graphical user interface.

Because of the offset between the actual ADC clock frequency f_(S)′ andthe intended clock frequency f_(S), the receiver's task of demodulatingthe received baseband signal is challenging and prone to large BERs. Inapplications involving the test and measurement of devices such as radiotransceivers (e.g. mobile phones), a transmitter generates a basebandsignal, and transmits the baseband signal. A signal analyzer receives anoise-perturbed version of the transmitted baseband signal, and analyzesthe received baseband signal, e.g., to evaluate the performance of thetransmitter. Error vector magnitude (EVM) is a widely used metric toevaluate the modulation quality of the transmitter.

In the presence of a clock frequency offset, one of the requirements ofthe measurement application is to estimate the clock frequency offset,or equivalently, the relative clock offset ε. This problem becomes muchmore complex in the presence of carrier frequency offset and carrierphase offset. Carrier frequency offset is the difference between thefrequency of the receiver's carrier signal and the frequency of thetransmitter's carrier signal. Similarly, carrier phase offset is thedifference between the phase of the receiver's carrier signal and thephase of the transmitter's carrier signal.

Signal Model

At the transmitter, a stream of information bits {b_(k)} is mapped intoa stream of complex symbols {U_(n)} in a constellation (i.e., a set ofpoints in the complex plane), e.g., a QAM constellation. The stream ofsymbols {U_(n)} are used to generate a stream of samples. (The number ofsamples per symbol is referred to as the SPS factor.) The samples aresupplied to a subsystem that converts the samples into an analogbaseband signal u(t) based on a conversion clock. The analog basebandsignal u(t) may be used to modulate a carrier signal (or an orthogonalpair of carrier signals). The modulated carrier is transmitted through achannel, e.g., a wired channel such as a cable, a wireless channel suchas the atmosphere or free space, a fiber optic channel, etc. A receivercaptures a channel-distorted version of the transmitted signal from thechannel, recovers a baseband signal from the captured signal, andsamples the baseband signal. However, the sampling clock rate 1/T_(S)′used by the receiver is in general different from an intended samplingclock rate 1/T_(S).

Let X(t)=A(t)exp(jα(t)) denote the received baseband signal, where A(t)is the instantaneous amplitude and α(t) is the instantaneous phase.Thus, the samples {X(nT_(S)′)} captured by the receiver's ADC may berepresented as:X(nT′ _(S))=A(nT′ _(S))exp(jα(nT′ _(S)))  (2)

Ideally, the clock period T_(S)′ would equal the intended period T_(S),and the receiver's symbol-timing recovery would be perfect. In thiscase, the received samples would be:X(nT _(S))=A(nT _(S))exp(jα(nT _(S)))  (3)

With appropriate normalization of signal power, the samples {X(nT_(S))}might be taken as (or used to determine) estimates of the transmittedsymbols {U_(n)}. In general, however, the clock period T_(S)′ is notequal to intended clock period T_(S). Thus, the receiver may operate onthe received samples {X(nT_(S)′)} in order to estimate the relativeclock period offset ε. The estimated value of ε (or a graphicalrepresentation thereof) may be displayed to a user of the receiver.Furthermore, the receiver may compute an estimate of the clock periodT_(S)′ based on the estimated value of ε and the intended period valueT_(S).

In some embodiments, the estimated value of ε may be used tofractionally resample the received samples in order to obtain correctedsamples that correspond to the intended clock frequency 1/T_(S). In onealternative embodiment, the receiver may use the estimated value of ε toadjust the sampling clock frequency f_(S)′=1/T_(S)′ of the receiver'sADC. The adjustment causes the sampling clock frequency f_(S)′ toconverge towards the transmitter's conversion clock frequencyf_(S)=1/T_(S).

Expression (2) for the received samples assumes that there is no carrierfrequency offset and no carrier phase offset in the receiver, i.e., thatthe receiver's carrier frequency and phase exactly match thetransmitter's carrier frequency and phase, respectively. A more generalexpression for the received samples (after symbol timing recovery),admitting the possibility of both carrier frequency offset and carrierphase offset, looks like:X(nT′ _(S))=A(nT′ _(S))exp{j(α(nT′ _(S))+θ₀+Δω₀ nT′ _(S))}  (4)where Δω₀ is the carrier frequency offset in radians per sample, and θ₀is the angle error corresponding to the carrier phase offset.

From expression (1), it follows that T′_(S)=(1+ε)T_(S). Thus, expression(4) may be equivalently written as:X((1+ε)nT _(S))=A((1+ε)nT _(S))exp(jφ(n))  (5A)φ(n)=α((1+ε)nT _(S))+θ₀+Δω₀(1+ε)nT _(S)  (5B)

Algorithm for Estimating Relative Clock Period Offset

An error vector signal e(n) is computed as the difference between anintermediate signal Y(n) and a reference signal r(n):e(n)=Y(n)−r(n)  (6)(The quantity e(n) is a complex number, or equivalently, a vector in thecomplex plane. Thus, the term “vector” is used to describe e(n), and thesignal {e(n)} is said to be a “vector signal”.)

The reference signal r(n) represents an initial estimate for the idealsamples {X(nT_(S))}. (The receiver may use the reference signal r(n) andthe intermediate signal Y(n) to determine an improved estimate for theideal samples as described hereinafter. The computation of the errorvector signal e(n) is a step in that determination.) In other words,r(n)≈X(nT _(S))=A(nT _(S))exp(jα(nT _(S)))  (7)(Note that the relative clock offset ε is not present in the referencesignal r(n), having been eliminated by the hard decision demodulation.)

The samples {r(n)} of the reference signal may be computed from thereceived samples {X(nT′_(S))} as follows. First, methods for removingcarrier frequency offset (CFO) and carrier phase offset (CPO) areapplied to the received samples {X(nT′_(S))} to obtain correctedsamples. Those methods may rely on a preamble (or a set of pilotsymbols) that has been embedded in the transmitted signal by thetransmitter. Such methods are well known to those of ordinary skill inthe art of signal processing. For more information on the methods forremoving CFO and CPO, please see the following references: (a) U.Mengali and A. N. D'Andrea, “Synchronization Techniques for DigitalReceivers”, New York, Plenum, 1997; (b) “WCDMA for UMTS”, edited by H.Holma and A. Toskala, Wiley, New York, 2003; and (c) M. Luise and R.Reggiannini, “Carrier Frequency Recovery in All-Digital Modems forBurst-Mode Transmissions,” IEEE Transactions on Communications, Vol. 43,No. 234, pages 1169-1178, February-April 1995.

After removal of the CFO and CPO, the corrected samples may be subjectedto hard decision demodulation in order to obtain a sequence of bitvalues. The sequence of bit values resulting from the hard decisiondemodulation may then be re-modulated to obtain the samples {r(n)} ofthe reference signal. The process of hard decision demodulation is wellknown to those of ordinary skill in the art of signal processing. Formore information on hard decision demodulation, please see either of thefollowing references: (1) “Using Error Vector Magnitude Measurement toAnalyze and Troubleshoot Vector Modulate Signals”, Agilent Product Note89400-14, Agilent Technologies Literature No. 5965-2898E, 2000; (2)Voelker, Kenneth M. “Apply Error Vector Measurements in CommunicationsDesign”, Microwaves & RF, December 1995, pp. 143-152.

The constellation of the reference signal {r(n)} is scaled to theaverage power of the received signal {X(nT_(S)′)}.

As shown in expression (6) above, the computation of the error vectorsignal e(n) also involves the intermediate signal Y(n). The samples{Y(n)} of the intermediate signal are computed from the received samples{X(nT′_(S))} by removing the CFO and CPO with the help of the referencesignal. In one embodiment, an estimate for the carrier frequency offsetΔω₀ may be determined by computing the average slope over N samples ofthe sequence {arg(X(nT_(S)′)r(n)*)}, and, an estimate for the carrierphase offset θ₀ may be determined by computing a phase intercept for thesame sequence (interpreted as a function of index n). The notationarg(z) represents the phase (angle) of the complex number z. Thesuperscript “*” denotes complex conjugation.

The samples {Y(n)} may be represented by the expression:Y(n)=A(nT′ _(S))exp(jα(nT′ _(S)))  (8)Expression (8) assumes that there is no CFO or CPO in the intermediatesignal Y(n). However, because of the sample clock offset ε, the CFOremoval algorithm and the CPO removal algorithm may fail to completelyremove the CFO and CPO from the received samples. As a result there maybe some small residual CFO and residual CPO in the intermediate signalY(n). In this case the intermediate signal Y(n) would have the form:Y(n)=A(nT′ _(S))exp{j(α(nT′ _(S))+Δθ+Δω′₀ nT′ _(S))}  (9)where Δω′₀ is the residual CFO, and Δθ is the residual CPO.

In the discussion below, an expression for the absolute value |e(n)| ofthe error vector e(n) will be developed based on the intermediate signalY(n) as given by equation (8). (The absolute value |z| of any complexnumber z=a+ib is defined as |z|=√{square root over (a²+b²)}.)Furthermore, an expression for the absolute value |e′(n)| of adifferential error vector e′(n) will be developed based on theintermediate signal Y(n) as given by expression (9). In each case, aformula will be established for the relative clock offset ε in terms ofthe corresponding absolute value data. Either of those formulas may beused to estimate ε. (The first formula, based on the absolute value|e(n)|, may be used, e.g., in situations where the residual CFO and theresidual CPO are believed to be negligibly small. Otherwise, the latterformula, based on the absolute value |e′(n)|, may be used.) Theestimated value of ε may then be used to fractionally resample thereceived samples {X(nT′_(S)′)} in order to arrive at a more perfectapproximation of the ideal samples {X(nT_(S))}, and thus, of thetransmitted symbol sequence.

Case-I: Assuming that the intermediate signal Y(n) has no CFO or CPO,the error vector e(n) may be written as:e(n)=A(nT′ _(S))exp(jα(nT′ _(S)))−A(nT _(S))exp(jα(nT _(S)))  (10)

This expression (10) may be simplified, by making the followingassumptions:

(A1) A(nT′_(S))=A(nT_(S)) for all n. This assumption is made because thedifference between A(nT_(S)) and A(nT_(S)′) is a significantly smallercontributor to error vector e(n) than is the difference betweenα(nT_(S)) and α(nT_(S)′).

(A2) The phase α(t) is assumed to be linear between any intendedsampling time nT_(S) and the corresponding actual sampling timenT_(S)′=(1+ε)nT_(S). This linearity is illustrated in FIG. 1.

(A3) The accumulated error in clock phase is low enough not to cause asignificant number of hard decision errors over the range of thereceived sample set.

Employing these assumptions, expression (10) may be rewritten as:

$\begin{matrix}{{{e(n)} = {2\;{A\left( {nT}_{S} \right)}{\sin\left( \frac{\beta(n)}{2} \right)}\exp\left\{ {j\left( {{\alpha\left( {nT}_{S} \right)} + \frac{\beta(n)}{2} + \frac{\pi}{2}} \right)} \right\}}},} & (11)\end{matrix}$where β(n)=α(nT′_(S))−α(nT_(S)). The absolute value of the error vectore(n) may be normalized to ensure that |e(n)| is in the range (0,1],e.g., by dividing equation (11) by |2A(nTs)|. The normalized absolutevalue of the error vector e(n) is then given by:

$\begin{matrix}{{{e(n)}} = {{{\sin\left( \frac{\beta(n)}{2} \right)}}.}} & (12)\end{matrix}$

It can be shown that β(n)≈Kεn, where K is the slope of the phasefunction θ(v) between v=n and v=(1+ε)n, where the phase function θ(v) isdefined by the relation:θ(v)=α(vT _(S)).

The value of slope K may be empirically estimated as discussed below.Thus, the expression (12) can be re-written as:

$\begin{matrix}{{{e(n)}} = {{{\sin\left( \frac{K\; ɛ\; n}{2} \right)}}.}} & (13)\end{matrix}$

When |ε| is small, expression (13) may be approximated by:

$\begin{matrix}{{{{e(n)}} = \frac{{{K\; ɛ}}n}{2}},} & (14)\end{matrix}$assuming the symbol index n is non-negative.

Observe that the absolute value function |e(n)| is linear in the sampleindex n, with slope given by:grad(|e(n)|)=|Kε|/2where “grad” denotes the first-order derivative with respective to indexn. Thus, the absolute value of the sample clock offset ε is given by:

$\begin{matrix}{{ɛ} = {\frac{2}{K}{{{grad}\left( {{e(n)}} \right)}.}}} & (15)\end{matrix}$

The slope grad(|e(n)|) may be computed using any of a variety ofnumerical differentiation formulas. For example, in one embodiment, theslope is computed based on the expression:grade(|e(n)|)=|e(n)|−|e(n−1)|  (16)

Expression (15) may be used by the receiver to compute an estimate for|ε|. However, to average out the affect of noise in the received signalsamples, the value |ε| may be computed using an average value of theslope grad(|e(n)|):

$\begin{matrix}{{ɛ} = {\frac{2}{K}{AVG}{\left\{ {{grad}\left( {{e(n)}} \right)} \right\}.}}} & (17)\end{matrix}$

In one embodiment, the average slope may be computed based on theexpression

$\begin{matrix}{{{{AVG}\left\{ {{grad}\left( {{e(n)}} \right)} \right\}} = \frac{\sum\limits_{n = 1}^{N - 1}\left( {{{e(n)}} - {{e\left( {n - 1} \right)}}} \right)}{N - 1}},} & (18)\end{matrix}$where N is the number of samples {X(nT′_(S))} in the received sampleset. Other methods for calculating the average slope are contemplated.For example, in one alternative embodiment, the average slope iscomputed based on a least squares fit of a line having the form y=mx tothe data set{(n,grad(|e(n)|)):n=0,1, . . . ,N−1}

To use expression (17) as a means for estimating |ε|, the receiver needsto have an estimate for the constant K. The value of constant K isdependent on the sampling period T_(S)′ and can be derived as part of acalibration procedure prior to operating on the received samples. Thecalibration procedure involves a stored set of baseband signal samples{S(nTs′)}. The samples {S(nTs′)} have been captured with a known valueof clock period Ts′, and stored into the memory of the receiver alongwith the known value of the clock period Ts′. In one embodiment, thesamples {S(nTs′)} may be captured in a factory using highly accurateequipment to measure the actual period Ts′ (or actual clock frequency1/Ts′) used by the receiver's ADC while it is capturing the samples{S(nTs′)}. The samples {S(nTs′)}, the known value of the clock periodTs′ and the intended period value T_(S) may be stored in the memory ofthe receiver, e.g., prior to shipping the receiver to a user/customer.

At calibration time, the receiver may access the samples {S(nTs′)} frommemory and perform the above-described computation of the functiongrad(|e(n)|) based on the samples {S(nTs′)}. Furthermore, the receivermay access the intended period value Ts and the known period value Ts′from the memory, and compute the parameter ε₀ according to the relation

$ɛ_{0} = {\frac{T_{S}^{\prime} - T_{S}}{T_{S}}.}$Finally, the receiver may estimate the absolute value of the constant Kbased on the relation:

$\begin{matrix}{{K} = {\frac{2\mspace{11mu}{AVG}\left\{ {{grad}\left( {{e(n)}} \right)} \right\}}{ɛ_{0}}.}} & (19)\end{matrix}$

Case-II: Assuming that the intermediate signal Y(n) has both residualCFO and residual CPO, the error vector e(n) may be written as:e(n)=Y(n)−r(n)  (20A)e(n)=A(nT′ _(S))exp{j(α(nT′ _(S))+Δθ+Δω′₀ nT′ _(S))}−A(nT _(S))exp{jα(nT_(S))}  (20B)Observe that this expression involves three unknowns: T_(S)′, Δω₀′ andΔθ (or equivalently, ε, Δω₀′ and Δθ).

It is a fundamental fact that by performing the differential of anysignal {v(n)}, the CFO in the signal {v(n)} will manifest as a constantphase offset error in the differential signal, and the phase offsetpresent in the signal {v(n)} will be removed from the differentialsignal. The differential signal ∇v(n) is defined as:∇v(n)=v(n)v(n−1)*  (21)

The error vector e′(n) corresponding to the differential of theintermediate signal Y(n) and the differential of the reference signalr(n) is given by:e′(n)=∇Y(n)−∇r(n)  (22)

Applying the definition of differential to the intermediate signal Y(n),one obtains:∇Y(n)=A(nT′ _(S))A((n−1)T′ _(S))exp{j(α(nT′ _(S))−α((n−1)T′ _(S))−Δω′₀T′ _(S))}  (23)Observe that term involving the residual carrier frequency offset Δω₀′appears as a constant phase offset, whereas the corresponding term inexpression (9) was linear in index n.

Similarly, the differential of the reference signal r(n) is given by:∇r(n)=A(nT _(S))A((n−1)T _(S))exp{j(α(nT _(S))−α((n−1)T _(S)))}  (24)

Substituting (23) and (24) into (22) and applying the same assumptionsas used in Case I, one obtains:

$\begin{matrix}{{e^{\prime}(n)} = {2{B(n)}\exp\left\{ {j\;{\psi(n)}} \right\}{{\sin\left( \frac{{\beta(n)} - {\beta\left( {n - 1} \right)}}{2} \right)}.}}} & \left( {25A} \right) \\{{\psi(n)} = {{\alpha\left( {nT}_{S} \right)} - {\alpha\left( {\left( {n - 1} \right)T_{S}} \right)} + \frac{{\beta(n)} - {\beta\left( {n - 1} \right)}}{2} + \frac{\pi}{2}}} & \left( {25B} \right)\end{matrix}$

Thus, the absolute value of the error vector e′(n) after normalizationis given by:

$\begin{matrix}{{{e^{\prime}(n)}} = {{{\sin\left( \frac{{\beta(n)} - {\beta\left( {n - 1} \right)}}{2} \right)}}.}} & (26)\end{matrix}$

Using the assumption (A2) described above in Case I, expression (26) maybe written as:

$\begin{matrix}{{{e^{\prime}(n)}} = {{{\sin\left( \frac{K\; ɛ}{2} \right)}}.}} & (27)\end{matrix}$(Recall that β(n)≈Kεn. But this implies β(n)−β(n−1)≈Kεn−Kε(n−1)=Kε.) Andthus, under the condition that |ε| is small, one can write:

${{{e^{\prime}(n)}} = \frac{{K\; ɛ}}{2}},$or equivalently,

$\begin{matrix}{{ɛ} = {\frac{2{{e^{\prime}(n)}}}{K}.}} & (28)\end{matrix}$

The receiver may estimate |ε| according to the relation:

$\begin{matrix}{{{ɛ} = \frac{2R\;{MS}\left\{ {{e^{\prime}(n)}} \right\}}{K}},} & (29)\end{matrix}$where the notation RMS {v(n)} denotes the RMS value of the sequence{v(n)}, i.e.,

${R\;{MS}\left\{ {v(n)} \right\}} = {\sqrt{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{v(n)}}^{2}}}.}$

The value of the constant K may be derived by calibration, e.g., acalibration performed prior to operating on the received samples. Thecalibration procedure involves a set of baseband signal samples{S(nTs′)} that have been stored into the memory of the receiver, asdescribed above. At calibration time, the receiver may access thesamples {S(nTs′)} from memory, and perform the above-describedcomputation of the value RMS{|e′(n)|} based on the samples {S(nTs′)}.Furthermore, the receiver may access the intended period value Ts andthe known period value Ts′ from the memory, and compute the parameter ε₀according to the formula:

$ɛ_{0} = {\frac{T_{S}^{\prime} - T_{S}}{T_{S}}.}$

Finally, the receiver may estimate the absolute value of the constant Kbased on the relation:

${K} = {\frac{2\; R\;{MS}\left\{ {{e^{\prime}(n)}} \right\}}{ɛ_{0}}.}$

Resolving Sign Ambiguity

As noted above, either formula (17) or formula (29) may be used tocompute the estimate for |ε|. To determine the sign of ε, the followingprocedure may be used. First, the receiver fractionally resamples thereceived baseband samples {x(nT_(S)′)} by a factor of (1+|ε|) to obtaina first resampled signal {Z₁(n)} corresponding to the ratef₁=(1+|ε|)/T_(S)′; and fractionally resamples the received basebandsamples {x(nT_(S)′)} by a factor of (1−|ε|) to obtain a second resampledsignal {Z₂(n)} corresponding to the rate f₂=(1−|ε|)/T_(S)′. Resamplingby a factor p means the ratio of the number of output samples to thenumber of input samples is equal to p. (For example, if p=1001/1005, theresampling would produce 1001 output samples for every 1005 inputsamples.) (Note that the value of clock period T_(S)′ is as yet unknown.Thus, f₁ and f₂ are not yet known.)

Second, the receiver computes two error vector signals according to theexpressions:e ₁(n)=Z ₁(n)−r(n)  (30A)e ₁(n)=Z ₂(n)−r(n)  (30B)

Third, the receiver computes two RMS values according to theexpressions:R ₁ =RMS{|e ₁(n)|}  (31A)R ₂ =RMS{|e ₂(n)|}  (31B)

The smaller of the two RMS values determines the sign of ε. In otherwords, if R₁ is smaller than R₂, the receiver sets the sign of ε to bepositive, performs demodulation on the samples {Z₁(n)}, and discards thesamples {Z₂(n)}. On the other hand, if R₂ is smaller than R₁, thereceiver sets the sign of ε to be negative, performs demodulation on thesamples {Z₂(n)}, and discards the samples {Z₁(n)}. The process ofdemodulation produces a sequence of bits that represent estimates of theoriginal information bits {b_(k)}.

Methods for performing fractional resampling are well known to those ofordinary skill in the art of signal processing. For information onmethods for performing fractional resampling, see, e.g., “MultirateSystems And Filter Banks” by P. P. Vaidyanathan, Prentice Hall PTR 1992(ISBN: 0136057187).

In some embodiments, the receiver is a signal analyzer. In oneembodiment, the signal analyzer includes one or more devicesmanufactured by National Instruments Corporation. For example, in oneembodiment, the signal analyzer is the PXIe-5663E 6.6 Ghz RF VectorSignal Analyzer.

In one set of embodiments, a method for operating a receiver system mayinvolve the operations shown in FIG. 2.

At 210, an analog-to-digital converter (ADC) of the receiver system maycapture samples {x(nT_(S)′)} of a received baseband signal, where n isthe sample index, where T_(S)′ is the period of the ADC's sample clock.The receiver system may include hardware for receiving a signal from atransmission channel, and for down-converting the received signal tobaseband. The baseband signal is provided to the ADC, which captures thesamples of the baseband signal based on the sample clock. The actualperiod T_(S)′ of the sample clock is assumed to be unknown to thereceiver system, at least initially. (After estimating the value of ε bythe present method, however, the receiver system may use that estimateto compute a value for the period T_(S)′.)

At 215, a processor of the receiver system may compute a referencesignal {r(n)} by operating on the baseband signal samples {x(nT_(S)′)}to perform carrier-frequency-offset (CFO) removal, carrier-phase-offset(CPO) removal and hard decision demodulation in order to obtain a bitstream, and by remodulating the bit stream to determine by the referencesignal, e.g., as variously described above.

At 220, the processor may compute an intermediate signal {Y(n)} byoperating on the baseband signal samples {x(nT_(S)′)} to perform CFOremoval and CPO removal using the reference signal {r(n)}, e.g., asvariously described above.

At 225, the processor may compute an error vector signal {e(n)} based onthe intermediate signal {Y(n)} and the reference signal {r(n)}, e.g.,according to the relation:e(n)=Y(n)−r(n)

At 230, the processor may compute an estimate Q for the absolute value|ε| of an offset parameter ε based on an average slope of an absolutevalue of the error vector signal {e(n)}, e.g., as variously describedabove. The offset parameter ε represents a relative offset between theperiod T_(S)′ and an intended period T_(S) of the ADC's sample clock.

At 235, the processor may fractionally resample the baseband signalsamples {x(nT_(S)′)} by a factor of (1+Q) to obtain a first set ofadjusted samples {Z₁(n)} corresponding to sample rate f₁=(1+Q)/T_(S)′,and fractionally resample the baseband signal samples {x(nT_(S)′)} by afactor of (1−Q) to obtain a second set of adjusted samples {Z₂(n)}corresponding to sample rate f₂=(1−Q)/T_(S)′, e.g., as variouslydescribed above.

At 240, the processor may compute a first error vector signal {e₁(n)}based on the first set of adjusted samples {Z₁(n)} and the referencesignal {r(n)}, and compute a second error vector signal {e₂(n)} based onthe second set of adjusted samples {Z₂(n)} and the reference signal{r(n)}, e.g., as described above.

At 245, the processor may select either Q or −Q as an estimate{circumflex over (ε)} for the offset parameter ε, where the selection ofeither Q or −Q is based on whether or not an RMS value R₁ of the firsterror vector signal {e₁(n)} is smaller than an RMS value R₂ of thesecond error vector signal {e₂(n)}. In other words, if R₁ is smallerthan R₂, the value Q is selected as the estimate for ε; otherwise thevalue −Q is selected.

After having selected the estimate {circumflex over (ε)} for the offsetparameter ε, that estimate may be stored in memory of the receiver,and/or, displayed to a user via a display device. In some embodiments,the above method may be performed repeatedly for a succession of blocksof the received signal samples. The sequence of estimates {circumflexover (ε)}(k) thereby generated may be graphed (or otherwise displayed)to give the user a sense of the time-variation of the ADC's sample clockperiod (or frequency).

In some embodiments, the processor may output (e.g., display) either thevalue 1/((1+Q)T_(S)) or the value 1/((1−Q)T_(S)) as an estimate of thefrequency 1/T_(S)′ of the ADC's sample clock based on whether or not theRMS value R₁ of the first error vector signal is smaller than the RMSvalue R₂ of the second error vector signal. In other words, if R₁ issmaller than R₂, the value 1/((1+Q)T_(S)) is outputted; otherwise thevalue 1/((1−Q)T_(S)) is outputted. The intended period value T_(S) isassumed to be a known parameter, e.g., known by virtue of thecommunication protocol being employed and the SPS factor being used bythe receiver. Thus, the values 1/((1+Q)T_(S)) and 1/((1−Q)T_(S)) may becomputed once the value Q has been computed.

In some embodiments, the processor may select either the first set ofadjusted samples {Z₁(n)} or the second set of adjusted samples {Z₂(n)}for output based on whether or not the RMS value R₁ of the first errorvector signal is smaller than the RMS value R₂ of the second errorvector signal. In other words, the first set of adjusted samples {Z₁(n)}is selected if R₁ is smaller than R₂; otherwise the second set ofadjusted samples {Z₂(n)} is selected. The selected set of adjustedsamples represents a resampling of the samples {x(nT_(S)′)} to theintended sample rate 1/T_(S). The non-selected set of samples may bediscarded.

The processor may output the selected set of adjusted samples. Theprocess of outputting the selected set of adjusted samples may involvestoring the selected set of adjusted samples in an output buffer, e.g.,a buffer that is accessible by a demodulation process (or some othercomputational process) that executes as part of the receiver system oras part of some system/device external to the receiver system.

In some embodiments, the processor (or some other processing agent) maydemodulate the selected set of adjusted samples in order to recoverinformation bits. (The demodulation corresponds to the type ofmodulation used by the transmitter, e.g., QAM in one embodiment.) Theinformation bits may be stored in memory, e.g., as part of an outputfile or output packet. In one embodiment, the processor may generate anoutput signal based on the information bits (e.g., an audio signaland/or a video signal), and provide the output signal to an outputdevice. The output device may be a display device and/or a set of one ormore speakers.

As described above, the processor computes (in operation 230) anestimate Q for the absolute value of the offset parameter ε based on anaverage slope of the absolute value of the error vector signal {e(n)}.In one embodiment, the average slope is computed according to theexpression:

${{{AVG}\left\{ {{grad}\left( {{e(n)}} \right)} \right\}} = \frac{\sum\limits_{n = 1}^{N - 1}\left( {{{e(n)}} - {{e\left( {n - 1} \right)}}} \right)}{N - 1}},$where N is the number of samples of the error vector signal {e(n)}.

In some embodiments, the smaller of the two values R₁ and R₂ may beoutputted as a measure of quality of the transmitter. (The receivedbaseband signal is a noise-perturbed version of a transmitted basebandsignal generated by a transmitter.)

In some embodiments, the receiver system is (or includes) a signalanalyzer. In one embodiment, the signal analyzer is realized by thePXIe-5663E 6.6 Ghz RF Vector Signal Analyzer provided by NationalInstruments Corporation.

In some embodiments, the receiver is a mobile telecommunication device,e.g., a mobile phone.

In some embodiments, the receiver is a network interface for a computersystem.

In one set of embodiments, a method for operating a receiver system mayinvolve the operations shown in FIG. 3.

At 310, an analog-to-digital converter (ADC) of the receiver system maycapture samples {x(nT_(S)′)} of a received baseband signal, where n is asample index, where T_(S)′ is a period of the sample clock used by theADC. The receiver system may include hardware for receiving a signalfrom a transmission channel, and for down-converting the signal tobaseband. The baseband signal is provided to the ADC, which captures thesamples of the baseband signal based on the sample clock. The actualperiod T_(S)′ of the sample clock is assumed to be unknown to thereceiver system, at least initially. (After estimating the value of ε bythe present method, however, the receiver may estimate the value ofperiod T_(S)′ using the known value of the intended period T_(S).)

At 315, a processor of the receiver system may compute a referencesignal {r(n)} by operating on the baseband signal samples {x(nT_(S)′)}to perform carrier-frequency-offset (CFO) removal, carrier-phase-offset(CPO) removal and hard decision demodulation in order to obtain a bitstream, and by remodulating the bit stream, e.g., as variously describedabove.

At 320, the processor may compute an intermediate signal {Y(n)} byoperating on the baseband signal samples {x(nT_(S)′)} to perform CFOremoval and CPO removal using the reference signal {r(n)}, e.g., asdescribed above.

At 325, the processor may compute a differential {∇r(n)} of thereference signal {r(n)} and a differential {∇Y(n)} of the intermediatesignal {Y(n)}, e.g., as described above. See expression (21) above for adefinition of the differential operator ∇.

At 330, the processor may compute an error vector signal {e′(n)} basedon the differential signal {∇Y(n)} and the differential signal {∇r(n)},e.g., according to the relation: e′(n)=∇Y(n)−∇r(n).

At 335, the processor may compute an estimate Q for the absolute value|ε| of an offset parameter ε based on an RMS value of the error vectorsignal {e′(n)}, e.g., as variously described above. The offset parameterε represents a relative offset between the period T_(S)′ and an intendedperiod T_(S) of the ADC's sample clock.

At 340, the processor may: fractionally resample the baseband signalsamples {x(nT_(S)′)} by a factor of (1+Q) to obtain a first set ofadjusted samples {Z₁(n)} corresponding to sample rate f₁=(1+Q)/T_(S)′;and fractionally resample the baseband signal samples {x(nT_(S)′)} by afactor of (1−Q) to obtain a second set of adjusted samples {Z₂(n)}corresponding to sample rate f₂=(1−Q)/T_(S)′. Techniques for performingfractional resampling are well known to those of ordinary skill in theart of signal processing, and thus, need not be explained here.

At 345, the processor may compute a first error vector signal {e₁(n)}based on the first set of adjusted samples {Z₁(n)} and the referencesignal {r(n)}, and compute a second error vector signal {e₂(n)} based onthe second set of adjusted samples {Z₂(n)} and the reference signal{r(n)}, e.g., as described above.

At 350, the processor selects either Q or −Q as an estimate for theoffset parameter ε, where the selection of either Q or −Q is based onwhether or not an RMS value R₁ of the first error vector signal {e₁(n)}is smaller than an RMS value R₂ of the second error vector signal{e₂(n)}. In other words, the value Q is selected if R₁ is smaller thanR₂; otherwise the value −Q is selected.

After having selected the estimate {circumflex over (ε)} for the offsetparameter ε, that estimate may be stored in memory of the receiver,and/or, displayed to a user via a display device.

In some embodiments, the processor may output (e.g., display) either thevalue 1/((1+Q)T_(S)) or the value 1/((1−Q)T_(S)) as an estimate of thefrequency 1/T_(S)′ of the ADC's sample clock based on whether or not theRMS value R₁ of the first error vector signal is smaller than the RMSvalue R₂ of the second error vector signal. In other words, the value1/((1+Q)T_(S)) is selected if R₁ is smaller than R₂; otherwise the value1/((1−Q)T_(S)) is selected.

In some embodiments, the processor may select either the first set ofadjusted samples {Z₁(n)} or the second set of adjusted samples {Z₂(n)}for output based on whether or not the RMS value R₁ of the first errorvector signal is smaller than the RMS value R₂ of the second errorvector signal. In other words, the first set of adjusted samples {Z₁(n)}is selected if R₁ is smaller than R₂; otherwise the second set ofadjusted samples {Z₂(n)} is selected. The selected set of adjustedsamples represents a resampling of the received baseband samples{x(nT_(S)′)} to the intended sample rate 1/T_(S). The non-selected setof samples may be discarded.

The processor may output the selected set of adjusted samples. Theprocess of outputting the selected set of adjusted samples may involvestoring the selected set of adjusted samples in an output buffer, e.g.,a buffer that is accessible by a demodulation process (or some othercomputational process) that executes as part of the receiver system oras part of some system/device external to the receiver system.

In some embodiments, the processor (or some other processing agent) maydemodulate the selected set of adjusted samples in order to recoverinformation bits. (The demodulation corresponds to the type ofmodulation used by the transmitter, e.g., QAM in one embodiment.) Theinformation bits may be stored in memory, e.g., as part of an outputfile or output packet. In one embodiment, the processor may generate anoutput signal based on the information bits (e.g., an audio signaland/or a video signal), and provide the output signal to an outputdevice. The output device may be a display device and/or a set of one ormore speakers.

In some embodiments, the smaller of the two values R₁ and R₂ may beoutputted as a measure of quality of the transmitter, i.e., thetransmitter that the receiver is in communication with. (The receivedbaseband signal is a noise-perturbed version of a transmitted basebandsignal generated by the transmitter.) In one embodiment, the transmitteris device (e.g., a mobile transmitter or transceiver device such as acell phone) that is being tested.

In some embodiments, the receiver system is (or includes) a signalanalyzer. In one embodiment the signal analyzer is realized by one ormore products provided by National Instruments Corporation, as describedabove.

In one set of embodiments, a method for operating a receiver system mayinvolve the following operations.

The method may involve a processor of the receiver system computing areference signal {r(n)} by operating on baseband signal samples{x(nT_(S)′)} to perform carrier-frequency-offset (CFO) removal,carrier-phase-offset (CPO) removal and hard decision demodulation inorder to obtain a bit stream, and by remodulating the bit stream, e.g.,as variously described above. The index n is a sample index, and T_(S)′is the period of the sample clock of the ADC.

The method may also involve the processor computing an intermediatesignal {Y(n)} by operating on the baseband signal samples {x(nT_(S)′)}to perform CFO removal and CPO removal using the reference signal{r(n)}, e.g., as variously described above.

The method may also involve the processor computing an error vectorsignal {e(n)} based on the intermediate signal {Y(n)} and the referencesignal {r(n)}. As described above, the error vector signal may becomputed by subtracting the reference signal from the intermediatesignal. In one embodiment, the error vector signal is computed bysubtracting the differential from the reference signal from thedifferential of the intermediate signal, as described above.

The method may also involve the processor computing an estimate Q forthe absolute value |ε| of an offset parameter ε based on the errorvector signal {e(n)}, e.g., as variously described above. The offsetparameter ε represents a relative offset between the period T_(S)′ andan intended period T_(S) of the sample clock of the ADC.

The method may also involve the processor fractionally resampling thebaseband signal samples {x(nT_(S)′)} by a factor of (1+Q) to obtain afirst set of adjusted samples {Z₁(n)}, and fractionally resampling thebaseband signal samples {x(nT_(S)′)} by a factor of (1−Q) to obtain asecond set of adjusted samples {Z₂(n)}.

The method may also involve the processor selecting either Q or −Q as anestimate for the offset parameter ε, wherein said selecting of either Qor −Q is based on whether or not an error vector signal {e₁(n)}corresponding to the first set of adjusted samples {Z₁(n)} has smallerenergy than an error vector signal {e₂(n)} corresponding to the secondset of adjusted samples {Z₂(n)}.

Simulation Results

In one embodiment, the above-described methods and a simulationframework for testing the methods are implemented using the LabVIEWgraphical programming language. (LabVIEW is a software product ofNational Instruments.) One embodiment of the simulation framework isshown in FIG. 4.

At 410, a stream of random bits is generated.

At 415, the bits are modulated using a given modulation scheme, e.g.,QPSK, 16QAM or 64QAM. The modulation generates a sequence of basebandsymbols {U_(n)} at rate 1/T_(S) with oversampling factor equal to one.(While the simulation framework of FIG. 4 uses an oversampling factorequal to one, the present inventions do not impose any constraint on theoversampling factor.) The modulated signal is used as the test inputsignal.

Given a value of the relative clock period offset ε, the modulatedsignal is fractionally resampled to symbol rate 1/T_(S)′, whereT_(S)′=(1+ε)T_(S). See block 420 of FIG. 4. The fractional resamplingproduces the baseband signal samples {x(nT_(S)′)}.

At 425, the method of FIG. 2 or the method of FIG. 3 is then performedon the baseband signal samples {x(nT_(S)′)} to obtain an estimateε_(est) for the relative clock period offset ε and to fractionallyresample the signal to the rate (1+ε_(est))/T_(S)′.

At 430, the estimate ε_(est) and the relative clock period offset ε maybe compared, e.g., by calculating their difference. Both values and thedifference may be stored in memory for later visualization and/oranalysis.

FIG. 5 is a table showing one configuration that was used for simulationwithin the above framework.

FIG. 6 shows the clock frequency offset (f_(S)′−f_(S)) introduced by thesimulation, and the corresponding estimated value of the clock frequencyoffset using the algorithm of Case 1. (These results are based on theconfiguration shown in FIG. 5.)

Using the above simulation framework with the configuration of FIG. 5,the sample clock frequency offset is simulated by varying the sampleclock frequency from −500 Hz to +500 Hz in steps of 1 Hz. In everyiteration, the clock frequency offset is estimated and the impairedsignal is corrected for the same by the algorithm of Case 1. The graphrepresenting the sampling rate (Hz) vs. iteration number is shown in theFIG. 7. Line-1 (the line with steeper slope) represents the sample clockrate used to sample the baseband signal under test. Line-2 (the linewith smaller average slope) represents the estimated sample clockfrequency using the algorithm of Case 1. The experiments were alsocarried out with different values of CFO and different values of CPO.

FIGS. 8A and 8B represent the baseband 64QAM signal constellation before(FIG. 8A) and after (FIG. 8A) correction of the clock frequency offsetusing the algorithm of Case 1. In particular, FIG. 8A shows theconstellation of the input 64QAM baseband signal with the followingimpairments: CFO=1 Khz, CPO=25 degrees, Sample clock frequencyoffset=−499 Hz. FIG. 8B shows the constellation of the 64QAM basebandsignal after having applied the clock frequency offset correction aswell as CFO and CPO correction.

FIGS. 9A and 9B shows the plot of the EVM function |e(n)| respectivelyfor clock frequency offset=1149 Hz and clock frequency offset=−326 Hz,but without CFO or CPO impairments. The intended sample clock frequencyis 3.84 Mhz. The length of the signal is 4096 symbols at an oversamplingfactor of 1. The signal is a 64QAM modulated signal. It can be seen fromFIG. 9A that the plot of |e(n)| agrees with the analytically-derivedexpression (13). Similarly, the plot of |e(n)| given in FIG. 9B forsmaller values of clock frequency offset agrees with theanalytically-derived expression (14).

FIG. 10 illustrates one embodiment of a computer system 1000 that may beused to perform any of the method embodiments described herein, or, anycombination of the method embodiments described herein, or any subset ofany of the method embodiments described herein, or, any combination ofsuch subsets.

Computer system 1000 may include a processing unit 1010, system memory1012, a set 1015 of one or more storage devices, a communication bus1020, a set 1025 of input devices, and a display system 1030.

System memory 1012 may include a set of semiconductor devices such asRAM devices (and perhaps also a set of ROM devices).

Storage devices 1015 may include any of various storage devices such asone or more memory media and/or memory access devices. For example,storage devices 1015 may include devices such as a CD/DVD-ROM drive, ahard disk, a magnetic disk drive, magnetic tape drives, etc.

Processing unit 1010 is configured to read and execute programinstructions, e.g., program instructions stored in system memory 1012and/or on one or more of the storage devices 1015. Processing unit 1010may couple to system memory 1012 through communication bus 1020 (orthrough a system of interconnected busses). The program instructionsconfigure the computer system 1000 to implement a method, e.g., any ofthe method embodiments described herein, or, any combination of themethod embodiments described herein, or, any subset of any of the methodembodiments described herein, or any combination of such subsets.

Processing unit 1010 may include one or more processors (e.g.,microprocessors).

One or more users may supply input to the computer system 1000 throughthe input devices 1025. Input devices 1025 may include devices such as akeyboard, a mouse, a touch-sensitive pad, a touch-sensitive screen, adrawing pad, a track ball, a light pen, a data glove, eye orientationand/or head orientation sensors, a microphone (or set of microphones),or any combination thereof.

The display system 530 may include any of a wide variety of displaydevices representing any of a wide variety of display technologies. Forexample, the display system may be a computer monitor, a head-mounteddisplay, a projector system, a volumetric display, or a combinationthereof. In some embodiments, the display system may include a pluralityof display devices. In one embodiment, the display system may include aprinter and/or a plotter.

In some embodiments, the computer system 1000 may include other devices,e.g., devices such as one or more graphics accelerators, one or morespeakers, a sound card, a video camera and a video card.

In some embodiments, computer system 1000 may include one or morecommunication devices 1035, e.g., a network interface card forinterfacing with a computer network. In one embodiment, such acommunication device may be used to receive a modulated signal andperform processing operations on the modulated signal. The communicationdevice may include an embedded processor (such as a microprocessor)configured to execute program instructions and/or a programmablehardware element (such as an FPGA). The communication devices 1035 arepreferably configured to operate under the control of the softwareexecuting on processor 1010.

In some embodiments, the computer system 1000 may be configured forcoupling to a data acquisition system 1040. The data acquisition system1040 is configured to receive analog inputs signals, to digitize theanalog input signals, and to process the digitized signals and/or tomake the digitized signals available to other processing agents of thecomputer system 1000, e.g., to processor 1010.

In some embodiments, the computer system 1000 may interface with areceiver 1050 configured for receiving signals (such as radio signals,RF signals, wireless LAN signals, optical signals, etc.),down-converting the received signals, and capturing samples thedown-converted signals. The receiver may process the captured samplesand/or make the captured samples available to other processing agents ofthe computer system 1000, e.g., to processor 1010. The receiver may alsoinclude an embedded processor (configured to execute programinstructions) and/or a programmable hardware element (such as an FPGA).Thus, the computational effort of processing the captured samples may bedivided in various ways between the receiver and processor 1010. Any ofthe method embodiments described herein may be performed entirely on thereceiver, entirely on the processor 1010, or partly in the receiver andpartly on the processor 1010.

The computer system may be configured with a software infrastructureincluding an operating system, and perhaps also, one or more graphicsAPIs (such as OpenGL®, Direct3D, Java 3D™)

In some embodiments, the computer system 1000 may be provided withsoftware tools allowing a user to build programs (e.g., graphicalprograms). The software tools may enable the user to specify a targetdevice on which a program is to be executed/realized. For example, theuser may specify that a program may be realized on a processor and/or aprogrammable hardware element of a peripheral device such as receiver1050 or one or more of the communication devices 1035. For example, thecomputer system 1000 may be provided with a copy of LabVIEW and/orLabVIEW FPGA, which are software products of National InstrumentsCorporation.

FIG. 11 illustrates one possible embodiment 1100 of computer system1000. Although the embodiments above have been described in considerabledetail, numerous variations and modifications will become apparent tothose skilled in the art once the above disclosure is fully appreciated.It is intended that the following claims be interpreted to embrace allsuch variations and modifications.

What is claimed is:
 1. A method for operating a receiver system, themethod comprising: receiving, at a processor of the receiver system,samples {x(nT_(S)′)} of a received baseband signal, wherein n is asample index, where T_(S)′ is a period of a sample clock used to acquirethe samples {x(nT_(S)′)}; computing, at the processor, an intermediatesignal {Y(n)} by operating on the baseband signal samples {x(nT_(S)′)}to perform carrier frequency offset (CFO) removal and carrier phaseoffset (CPO) removal using a reference signal {r(n)}; computing, at theprocessor, an error vector signal {e(n)} based on the intermediatesignal {Y(n)} and the reference signal {r(n)}; computing, at theprocessor, an estimate Q for the absolute value |ε| of an offsetparameter ε0 based on an average slope of an absolute value of the errorvector signal {e(n)}, wherein the offset parameter ε represents arelative offset between the period T_(S)′ and an intended period T_(S)of the sample clock; fractionally resampling, at the processor, thebaseband signal samples {x(nT_(S)′)} by a factor of (1+Q) to obtain afirst set of adjusted samples {Z₁(n)}, and fractionally resampling, atthe processor, the baseband signal samples {x(nT_(S)′)} by a factor of(1−Q) to obtain a second set of adjusted samples {Z₂(n)}; computing, atthe processor, a first error vector signal {e₁(n)} based on the firstset of adjusted samples {Z₁(n)} and the reference signal {r(n)}, andcomputing, at the processor, a second error vector signal {e₂(n)} basedon the second set of adjusted samples {Z₂(n)} and the reference signal{r(n)}; selecting, at the processor, either Q or −Q as an estimate forthe offset parameter ε, wherein said selecting of either Q or −Q isbased on whether or not a root mean square (RMS) value R₁ of the firsterror vector signal {e₁(n)} is smaller than an RMS value R₂ of thesecond error vector signal {e₂(n)}.
 2. The method of claim 1, furthercomprising: outputting, at the processor, either a value 1/((1+Q)T_(S))or a value 1/((1−Q)T_(S)) as an estimate of a frequency 1/T_(S)′ of thesample clock based on whether or not the RMS value R₁ of the first errorvector signal is smaller than the RMS value R₂ of the second errorvector signal.
 3. The method of claim 1, further comprising: selecting,at the processor, either the first set of adjusted samples {Z₁(n)} orthe second set of adjusted samples {Z₂(n)} for output based on whetheror not the RMS value R₁ of the first error vector signal is smaller thanthe RMS value R₂ of the second error vector signal, wherein the selectedset of adjusted samples represents a resampling of the samples{x(nT_(S)′)} to a sample rate 1/T_(S).
 4. The method of claim 3, furthercomprising: demodulating the selected set of adjusted samples to recoverinformation bits; generating an output signal based on the informationbits; and providing the output signal to an output device.
 5. The methodof claim 1, wherein the received baseband signal is a noise-perturbedversion of a transmitted baseband signal generated by a transmitter,wherein the method further comprises: outputting, at the processor, asmaller one of the values R₁ and R₂, wherein the smaller valuerepresents a measure of quality of the transmitter.
 6. A non-transitorycomputer-readable memory medium for operating a receiver system, whereinthe memory medium stores program instructions, wherein the programinstructions, when executed by a processor, cause the processor to:receive samples {x(nT_(S)′)} of a received baseband signal, wherein n isa sample index, where T_(S)′ is a period of a sample clock used toacquire the samples {x(nT_(S)′)}; compute an intermediate signal {Y(n)}by operating on the baseband signal samples {x(nT_(S)′)} to performcarrier frequency offset (CFO) removal and carrier phase offset (CPO)removal using a reference signal {r(n)}; compute an error vector signal{e(n)} based on the intermediate signal {Y(n)} and the reference signal{r(n)}; compute an estimate Q for the absolute value |ε| of an offsetparameter ε based on an average slope of an absolute value of the errorvector signal {e(n)}, wherein the offset parameter ε represents arelative offset between the period T_(S)′ and an intended period T_(S)of the sample clock; fractionally resample the baseband signal samples{x(nT_(S)′)} by a factor of (1+Q) to obtain a first set of adjustedsamples {Z₁(n)}, and fractionally resample the baseband signal samples{x(nT_(S)′)} by a factor of (1−Q) to obtain a second set of adjustedsamples {Z₂(n)}; compute a first error vector signal {e₁(n)} based onthe first set of adjusted samples {Z₁(n)} and the reference signal{r(n)}, and compute a second error vector signal {e₂(n)} based on thesecond set of adjusted samples {Z₂(n)} and the reference signal {r(n)};select either Q or −Q as an estimate for the offset parameter ε, whereinsaid selecting of either Q or −Q is based on whether or not a root meansquare (RMS) value R₁ of the first error vector signal is smaller thanan RMS value R₂ of the second error vector signal.
 7. The memory mediumof claim 6, wherein the program instructions, when executed by theprocessor, further cause the processor to: output either a value1/((1+Q)T_(S)) or a value 1/((1−Q)T_(S)) as an estimate of a frequency1/T_(S)′ of the sample clock based on whether or not the RMS value R₁ ofthe first error vector signal is smaller than the RMS value R₂ of thesecond error vector signal.
 8. The memory medium of claim 6, wherein theprogram instructions, when executed by the processor, further cause theprocessor to: select either the first set of adjusted samples {Z₁(n)} orthe second set of adjusted samples {Z₂(n)} for output based on whetheror not the RMS value R₁ of the first error vector signal is smaller thanthe RMS value R₂ of the second error vector signal, wherein the selectedset of adjusted samples represents a resampling of the samples{x(nT_(S)′)} to a sample rate 1/T_(S).
 9. The memory medium of claim 8,wherein the program instructions, when executed by the processor,further cause the processor to: demodulate the selected set of adjustedsamples to recover information bits; generate an output signal based onthe information bits; and provide the output signal to an output device.10. The memory medium of claim 6, wherein the received baseband signalis a noise-perturbed version of a transmitted baseband signal generatedby a transmitter, wherein the program instructions, when executed by theprocessor, further cause the processor to: output a smaller one of thevalues R₁ and R₂, wherein the smaller value represents a measure ofquality of the transmitter.
 11. A method for operating a receiversystem, the method comprising: receiving samples {x(nT_(S)′)} of areceived baseband signal, wherein n is a sample index, where T_(S)′ is aperiod of a sample clock used to acquire the samples {x(nT_(S)′)};computing, at the processor, an intermediate signal {Y(n)} by operatingon the baseband signal samples {x(nT_(S)′)} to perform carrier frequencyoffset (CFO) removal and carrier phase offset (CPO) removal using areference signal {r(n)}; computing, at the processor, a differential ofthe reference signal {r(n)} and a differential of the intermediatesignal {Y(n)}; computing, at the processor, an error vector signal{e′(n)} based on the differential of the intermediate signal {Y(n)} andthe differential of the reference signal {r(n)}; computing, at theprocessor, an estimate Q for the absolute value |ε| of an offsetparameter ε based on the error vector signal {e′(n)}, wherein the offsetparameter ε represents a relative offset between the period T_(S)′ andan intended period T_(S) of the sample clock; fractionally resampling,at the processor, the baseband signal samples {x(nT_(S)′)} by a factorof (1+Q) to obtain a first set of adjusted samples {Z₁(n)}, andfractionally resampling, at the processor, the baseband signal samples{x(nT_(S)′)} by a factor of (1−Q) to obtain a second set of adjustedsamples {Z₂(n)}; computing, at the processor, a first error vectorsignal {e₁(n)} based on the first set of adjusted samples {Z₁(n)} andthe reference signal {r(n)}, and computing, at the processor, a seconderror vector signal {e₂(n)} based on the second set of adjusted samples{Z₂(n)} and the reference signal {r(n)}; selecting, at the processor,either Q or −Q as an estimate for the offset parameter ε, wherein saidselecting of either Q or −Q is based on whether or not an RMS value R₁of the first error vector signal is smaller than an RMS value R₂ of thesecond error vector signal.
 12. The method of claim 11, furthercomprising: outputting, at the processor, either a value 1/((1+Q)T_(S))or a value 1/((1−Q)T_(S)) as an estimate of a frequency 1/T_(S)′ of thesample clock based on whether or not the RMS value R₁ of the first errorvector signal is smaller than the RMS value R₂ of the second errorvector signal.
 13. The method of claim 11, further comprising:selecting, at the processor, either the first set of adjusted samples{Z₁(n)} or the second set of adjusted samples {Z₂(n)} for output basedon whether or not the RMS value R₁ of the first error vector signal issmaller than the RMS value R₂ of the second error vector signal, whereinthe selected set of adjusted samples represents a resampling of thesamples {x(nT_(S)′)} to a sample rate 1/T_(S).
 14. The method of claim13, further comprising: demodulating the selected set of adjustedsamples to recover information bits; generating an output signal basedon the information bits; and providing the output signal to an outputdevice.
 15. The method of claim 11, wherein the received baseband signalis a noise-perturbed version of a transmitted baseband signal generatedby a transmitter, wherein the method further comprises: selecting, atthe processor, a smaller one of the values R₁ and R₂, wherein thesmaller value represents a measure of quality of the transmitter.
 16. Anon-transitory computer readable memory medium for operating a receiversystem, wherein the memory medium stores program instructions, whereinthe program instructions, when executed by a processor, cause theprocessor to: receive samples {x(nT_(S)′)} of a received basebandsignal, wherein n is a sample index, where T_(S)′ is a period of asample clock used to acquire the samples {x(nT_(S)′)}; compute anintermediate signal {Y(n)} by operating on the baseband signal samples{x(nT_(S)′)} to perform carrier frequency offset (CFO) removal andcarrier phase offset (CPO) removal using a reference signal {r(n)};compute a differential of the reference signal {r(n)} and a differentialof the intermediate signal {Y(n)}; compute an error vector signal{e′(n)} based on the differential of the intermediate signal {Y(n)} andthe differential of the reference signal {r(n)}; compute an estimate Qfor the absolute value |ε| of an offset parameter ε based on the errorvector signal {e′(n)}, wherein the offset parameter ε represents arelative offset between the period T_(S)′ and an intended period T_(S)of the sample clock; fractionally resample the baseband signal samples{x(nT_(S)′)} by a factor of (1+Q) to obtain a first set of adjustedsamples {Z₁(n)}, and fractionally resample the baseband signal samples{x(nT_(S)′)} by a factor of (1−Q) to obtain a second set of adjustedsamples {Z₂(n)}; compute a first error vector signal {e₁(n)} based onthe first set of adjusted samples {Z₁(n)} and the reference signal{r(n)}, and compute a second error vector signal {e₂(n)} based on thesecond set of adjusted samples {Z₂(n)} and the reference signal {r(n)};select either Q or −Q as an estimate for the offset parameter ε, whereinsaid selecting of either Q or −Q is based on whether or not an RMS valueR₁ of the first error vector signal is smaller than an RMS value R₂ ofthe second error vector signal.
 17. The memory medium of claim 16,wherein the program instructions, when executed by the processor,further cause the processor to: output either a value 1/((1+Q)T_(S)) ora value 1/((1−Q)T_(S)) as an estimate of a frequency 1/T_(S)′ of thesample clock based on whether or not the RMS value R₁ of the first errorvector signal is smaller than the RMS value R₂ of the second errorvector signal.
 18. The memory medium of claim 16, wherein the programinstructions, when executed by the processor, further cause theprocessor to: select either the first set of adjusted samples {Z₁(n)} orthe second set of adjusted samples {Z₂(n)} for output based on whetheror not the RMS value R₁ of the first error vector signal is smaller thanthe RMS value R₂ of the second error vector signal, wherein the selectedset of adjusted samples represents a resampling of the samples{x(nT_(S)′)} to a sample rate 1/T_(S).
 19. The memory medium of claim18, wherein the program instructions, when executed by the processor,further cause the processor to: demodulate the selected set of adjustedsamples to recover information bits; generate an output signal based onthe information bits; and provide the output signal to an output device.20. The memory medium of claim 16, wherein the received baseband signalis a noise-perturbed version of a transmitted baseband signal generatedby a transmitter, wherein the program instructions, when executed by theprocessor, further cause the processor to: select a smaller one of thevalues R₁ and R₂, wherein the smaller value represents a measure ofquality of the transmitter.